1.2.J.sagews

from sympy.physics.units import *
from sympy import *

def prnt(format,x):
    try:
        x = x.applyfunc(simplify)
    except:
        x = x.simplify()
    if format=="tex":
        pprint(latex(x, mat_str="bmatrix", mat_delim=""))
    else:
        pprint(x)

symbolic = true
symbolic = false

if symbolic:
    a, b, c = var("a, b, c", positive=True)
    F = var("F", positive=True)
    alpha, beta = var("alpha, beta", positive=True)
else:
    # quantities:
    a, b, c = 4*m, 4*m, 2*m
    F = 6*newton
    alpha, beta = 30*pi/180, 60*pi/180

ca = cos(alpha)
sa = sin(alpha)
cb = cos(beta)
sb = sin(beta)

pprint("\n--- a ------------------------------------")
print("\nrP:")
rP = Matrix([-a, b, c])
prnt("tex", rP)

print("\n||rP||:")
rPn = rP.norm()
prnt("tex", rPn)

print("\ne:")
e = rP/rP.norm()
prnt("tex", e)

pprint("\n--- b ------------------------------------")
print("\nf:")
f = Matrix([-sa*cb, ca*cb, sb])
prnt("tex", f)
prnt("tex", N(f,2))

pprint("\n--- c ------------------------------------")
print("\nF:")
F = F*f
prnt("tex", F/newton)
prnt("tex", N(F/newton,2))

pprint("\n--- d ------------------------------------")
print("\nFpa:")
Fpa = F.dot(e)*e
Fpa = Fpa.applyfunc(simplify)
prnt("tex", Fpa)
prnt("tex", N(Fpa/newton,2))

pprint("\n--- e ------------------------------------")
print("\n||Fpa||:")
Fpam = Fpa.norm()
prnt("tex", Fpam)
prnt("tex", N(Fpam/newton,2))

pprint("\n--- f ------------------------------------")
print("\nFpe:")
Fpe = F-Fpa
Fpe = Fpe.applyfunc(simplify)
prnt("tex", Fpe)
prnt("tex", N(Fpe/newton,2))

pprint("\n--- g ------------------------------------")
Fpem = Fpe.norm()
Fpem = Fpem.simplify()
prnt("tex", Fpem)
prnt("tex", N(Fpem/newton,2))

pprint("\n--- h ------------------------------------")
prnt("tex", Fpam/newton)
prnt("tex", Fpem/newton)

lhs = Fpam**2 + Fpem**2
rhs = F.norm()**2
pprint(lhs.equals(rhs))

--- a ------------------------------------ rP: \begin{bmatrix}- 4 m\\4 m\\2 m\end{bmatrix} ||rP||: 6 m e: \begin{bmatrix}- \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{bmatrix} --- b ------------------------------------ f: \begin{bmatrix}- \frac{1}{4}\\\frac{\sqrt{3}}{4}\\\frac{\sqrt{3}}{2}\end{bmatr ix} \begin{bmatrix}-0.25\\0.43\\0.87\end{bmatrix} --- c ------------------------------------ F: \begin{bmatrix}- \frac{3}{2}\\\frac{3 \sqrt{3}}{2}\\3 \sqrt{3}\end{bmatrix} \begin{bmatrix}-1.5\\2.6\\5.2\end{bmatrix} --- d ------------------------------------ Fpa: \begin{bmatrix}- \frac{2 kg m}{3 s^{2}} \left(1 + 2 \sqrt{3}\right)\\\frac{2 k g m}{3 s^{2}} \left(1 + 2 \sqrt{3}\right)\\\frac{kg m}{3 s^{2}} \left(1 + 2 \s qrt{3}\right)\end{bmatrix} \begin{bmatrix}-3.0\\3.0\\1.5\end{bmatrix} --- e ------------------------------------ ||Fpa||: \frac{kg m}{s^{2}} \left(1 + 2 \sqrt{3}\right) 4.5 --- f ------------------------------------ Fpe: \begin{bmatrix}\frac{kg m}{6 s^{2}} \left(-5 + 8 \sqrt{3}\right)\\\frac{kg m}{ 6 s^{2}} \left(-4 + \sqrt{3}\right)\\\frac{kg m}{3 s^{2}} \left(-1 + 7 \sqrt{3 }\right)\end{bmatrix} \begin{bmatrix}1.5\\-0.38\\3.7\end{bmatrix} --- g ------------------------------------ \frac{kg m}{s^{2}} \sqrt{- 4 \sqrt{3} + 23} 4.0 --- h ------------------------------------ 1 + 2 \sqrt{3} \sqrt{- 4 \sqrt{3} + 23} True